Real Analysis -- Real Numbers/Integers Theory

Let x be a real number, and let N be an integer ≥ 2. Prove that there exist integers P and Q such that: 1 ≤ q ≤ N and absolute value of [x-(P/Q)] < 1/(QN)

Covers, Series Convergence

Say the only tool given to you is a calculator which performs addition, subtraction, multiplication, and division. Let X=(see attached). Explain a practical way of computing X within an error of 10^8. Roughly how big is X?

Inner Product Space Convergence

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Show that 2 sequences are orthonormal

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Hilbert space/subspace

Please see the attached file for full problem description.